On the Relationship between the Wigner-Moyaland Bohm Approaches to Quantum Mechanics:

A Step to a more General Theory?

B. J. Hiley!.

TPRU, Birkbeck, University of London, Malet Street,London WC1E 7HX.

(date)

AbstractIn this paper we show that the formalism used by Bohm in his

approach to quantum mechanics is already contained in the earlierpaper by Moyal which forms the basis for the Wigner-Moyal approach.This shows, contrary to the usual perception, that there is a deeprelation between the two approaches. We suggest the relevance ofthis result to the more general problem of constructing a quantumgeometry.

1 Introduction

It is a pleasure to contribute to this issue as Je! Bub and I go back a longway. We were members of the Physics Department at Birkbeck Collegeat the same time as David Bohm was appointed to a Chair in TheoreticalPhysics. I had just completed my PhD in condensed matter physics underthe supervision of Cyril Domb and Michael Fisher and had been appointedto the lowly post of Assistant Lecturer, although who I was supposed toassist was never made clear. Je!, fresh from Johannesburg, signed on fora PhD under Bohm’s supervision. He was interested in the foundations ofquantum mechanics and I wanted to move out of condensed matter physicsto do something of a more fundamental nature. In a sense we were bothembarking on the similar journey at the same time.

!E-mail address b.hiley@bbk.ac.uk.

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Found.Phys.,40 (2009) 356-67.

Je! became interested in the Weiner-Segal approach [1] to quantum me-chanics, a topic that became an important element of his thesis. He alsoengaged in a lively debate on the existence of hidden variables with Jauchand Piron [2]. While such debates have produced useful results, at that timeI did not feel they would be very fruitful in taking us further in our questfor a deeper understanding of quantum phenomena.

During this period Roger Penrose was in the Mathematics Departmentat Birkbeck developing his ideas on twistors [3] and we used to hold weeklyseminars ranging over many diverse topics in the foundations of physics, allchallenging the conventional wisdom. During these seminars I was particu-larly fascinated by the notions of quantum space times that were beginningto be discussed at the time. I was particularly fascinated by John Wheeler’s“sum over three geometries” ideas that he was using to quantise gravity. Iremembering him lecturing in London at the time and he produced one ofhis inimitable drawings that showed a set of three geometries lined up on theshelves of a book case! In figure 1 I reproduce some of Wheeler’s menagerieof three geometries [4] that made quite an impression on me.

Figure 1: The multisheeted character of superspace.

It was the properties of the spinor that gave rise to the multi-sheetedcharacter of the manifold. Here Wheeler had in mind the Pauli spinor.On the other hand, Penrose was generalising the spin structure by addingthe twistor, a spin object that occurred in conformal geometry. Part ofhis twistor programme was to construct a quantum geometry in which thetwistor variables would form a generalised phase space in which the con-sequences of curved space-time could be made manifest [5]. In presentinghis ideas on the twistor, Penrose used a notation that was peppered with

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indices and I found it di"cult to understand how it all fitted together. Iteventually dawned on me that there was a beautiful general mathematicaltheory that lay behind these spin structures, namely the Cli!ord algebraswhere one could dispense with many of the indices, making life a little eas-ier to see a rich structure emerging. These algebras formed a hierarchy ofincreasing dimension in which the Pauli spinor, the Dirac spinor and thetwistor appeared at di!erent levels in the hierarchy [6].

However all these algebras describe a classical geometry and it is not clearwhere the quantum aspects would come in. Wheeler did try to connect to thequantum domain by first starting with the classical Hamilton-Jacobi theoryand then trying to find what he called a “Einstein-Schrodinger equation”, asdid Penrose within his twistor phase space, However both met with limitedsuccess [7]. Nevertheless I felt that there was something compelling aboutthe whole approach.

One of the remarkable things about the sixties at Birkbeck was that, asfar as I recall, we never discussed in any detail Bohm’s original 1952 paper [8]that forms the basis of what later became known as “Bohmian mechanics”[9],[10]. Bohm himself was looking for a more fundamental structure ofideas upon which to base a new quantum theory. He was more interestedin developing a process based philosophy in which new forms of order andstructure would play a dominant role. It was, in fact, the beginning of whathe eventually called the implicate order [11]. Here Bohm proposed that themathematics best suited to describe order was the algebra and we exploredsome aspects of Cli!ord algebras hoping to find how from his general notionof order, space-time itself would arise [12]. This ‘pre-space’ order wouldform the basis of what people were calling quantum space-time [13].

All of this seemed a long way from the original ideas lying behind the ’52paper. This was why the Bohm model did not feature in our discussions.However when my attention was finally drawn to this paper, I suddenlyrealised that here was a possibility of finding clues to further Wheeler’sideas. The approach provided us with ‘geodesics’ and it also provided uswith a quantum Hamilton-Jacobi equation. Could a detailed study of thismodel provide the clues necessary to find the elusive “Einstein-Schrodingerequation”? It was this question that motivated me to become involved ina detailed study of the full implications of the Bohm model. Some of theresults of these investigations were presented in the book, “The UndividedUniverse”, that I co-authored with Bohm [14].

In this paper I want to report some new work that continues this line ofinvestigation. At first sight it might appear that it has little direct relevanceto the understanding of ‘quantum space-times’. However what I have come

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to realise is that to supply the missing quantum aspects to the classicalgeometric structures of space-time we need to investigate quantum phasespaces. But surely, you protest, the construction of a phase space is thevery thing that quantum mechanics denies? Seemingly not. Both the Bohmapproach [14] and the Wigner-Moyal approach [23] introduce phase spacesand yet both produce exactly the same statistical predictions as standardquantum mechanics. There is an extensive literature on both topics yet verylittle is written on the relation between the two phase spaces constructedin each case. I will spell out the relationship in this paper and discuss theimplications for the more general programme.

2 Two phase space approaches?

Over the years the original motivations for the work of Wigner [15], Moyal[16] and Bohm [8] seem to have been forgotten and their work is oftenpresented as attempts to revert to a ‘classical conception of nature’, yeteach paper contains clear warnings that their work does not imply a returnto classical physics. In the case of Bohm, this point is, perhaps, brought outin more detail in his book “Chance and Causality in Modern Physics” [17].

Wigner [15] was interested in quantum corrections to the Boltzmann dis-tribution used in thermodynamic equilibrium and involved many particles.However as he pointed out in the paper, it is possible to use the same for-malism to account for the statistical properties of single quantum systemsdescribed by a pure state wave function. Even in this case we have a functiongiven by

F (x, p) =1!

! "

#"!!(x! y)e2ipy/!!(x + y)dy (1)

and this can play the role of a ‘probability distribution function’ even thoughit takes negative values. Using it we can calculate the same expectationvalues as those calculated using the standard Hilbert space approach. Wedo this by simply integrating the operator equivalents over the position andmomentum as one does in classical physics (see equation (8)).

Moyal’s approach [16] starts by asking whether the standard formalismmight not disguise what is essentially a generalised statistical theory, but onethat is not necessarily based on a “crypto-deterministic” theory. He startsby taking a pair of non-commuting operators. (In this discussion we use theoperators p and x with eigenvalues p and x to illustrate the method.) Hethen defines M(", #) to be the expectation value of the Heisenberg operator

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ei(! p+"x). In statistics this object looks like the operator equivalent of thecharacteristic function of the distribution. In the state !(x), M(", #) takesthe form

M(", #) =! "

!!(x), ei(! p+"x)!(x)#

dx (2)

where the RHS is the standard inner product. Then the Fourier transformof M(", #) is

F (x, p) =1

4$2

! !M(", #)e#i(!p+"x)d"d# (3)

In turn this expression can then be written in the form [16]

F (x, p) =12$

!!!(x! !"/2)ei!p!(x + !"/2)d" (4)

which can be taken as a quasi probability distribution function. Apart froma trivial change of variable, this is identical to Wigner’s original expression(1).

After showing how the characteristic function can be used to reproduceall quantum expectation values, Moyal goes on to discuss the implication ofthis approach. He notes that the distributions can take on negative valuesand concludes, therefore, that not all quantum states can be represented bytrue probability distributions. Therefore this cannot be regarded as a classi-cal statistical theory. Nevertheless the existence of negative probabilities donot reveal themselves directly in any experiment and thus, from the practi-cal point of view, we can use these functions to aide our calculations, a pointof view that was originally suggested by Wigner [15] and later re-enforcedby Feynman [18].

Bohm’s approach [8] on the other hand was not motivated by an attemptto explain quantum mechanics in terms of a purely statistical theory. Ratherhe was concerned by the fact that there seemed to be no way in the conven-tional formalism to discuss what was happening between experiments. Inother words there was no way of talking about a possible objective processthat must be going on between measurements. The standard formalism im-plied that one could only talk about what happened when a measurementwas made and there was no way to talk about what the particle may bedoing between successive measurements. All we could do was to discussthe evolution of the wave function. For this reason the standard approachdiscourages any speculation as to what may be going on between measure-ments, believing we had reached the limits of the possibility of analysis interms of particle trajectories etc.

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However in the WKB approximation, we can talk about what the particlecould be doing between experiments. We can do this simply by neglectingterms of higher order than ! in the resulting series expansion. But thisprocess of approximation should not change the way we looked at the for-malism as a whole. Yet here we have started with the quantum mechanicalformalism with all its ambiguities and somehow “slipped over” into what isin essence, the ordinary classical ontology. How can simply throwing awayterms in the WKB expansion enable one to go from ambiguity to what ap-pears to be a deterministic theory? Why not keep all the terms and see whathappens to the classical description? Will there be any contradictions? Willwe learn anything new? It is not a question of forcing a classical view on theworld, rather it is a question of seeing what is the exact di!erence betweenclassical and quantum physics once both theories are reduced to the samemathematical language.

From the above remarks there seems on the surface to be no connectionbetween the Wigner-Moyal and Bohm approaches. Both start from verydi!erent outlooks and what also appears to be a very di!erent mathematics.Moyal’s approach assumes an underlying stochastic process in a generalisedphase space, the statistical distribution of which takes the form describedby equation (3).

Bohm on the other hand starts from the Schrodinger equation itself andchooses to write the wave function in the polar form ! = ReiS/!. Theresulting real part of the equation then gives

%S

%t+

12m

("S)2 + V ! !2

2m

"2R

R= 0 (5)

This is then treated as a modified Hamilton-Jacobi equation describing theevolution of a single quantum particle following one of the trajectories de-fined by the equation. It is this equation that we were referring to in theintroduction when we discussed Wheeler’s attempts to construct what hecalled the Einstein-Schrodinger equation. In our model we then simply as-sume that a particle can follow one of the ensemble of trajectories definedby the canonical relation

pB = "S (6)

These then would correspond to the geodesics in the Wheeler approach [4].Which particular trajectory a given particle travels along is unknown anduncontrollable since it is not possible to produce experimentally a singleparticle with a known simultaneous initial position and momentum. Thuswe are unable to produce a single particle travelling down a chosen pre-determined trajectory. All we are able to do is to produce a distribution

6

of possible initial conditions and therefore possible trajectories. This is thecontingent feature of the theory that gives rise to the statistics. The prob-ability is then obtained by averaging over all initial positions assuming thedistribution is consistent with the initial wave function. This final probabil-ity will always agree with the standard interpretation because the imaginarypart of the Schrodinger equation gives the conservation of probability

%&

%t+".

" &

m"S

#= 0 (7)

where &(x) is the usual probability density &(x) = !!(x)!(x).Again from these comments, it looks as if the two approaches are very

di!erent (see for example de Gosson [19]). Indeed, in certain circles, theWigner-Moyal approach is regarded as a valid approach of exploring particlebehaviour, whereas the Bohm approach is often dismissed as ‘meaningless’[20]. However what I want to bring out in this paper is that the equationsBohm uses for his approach already appear in the appendix of Moyal’s orig-inal 1949 paper [16]. This means that Bohm’s formalism already sits withinthe Wigner-Moyal formalism. In this paper I want to show exactly how theyare related.

3 The Common Momentum.

As we have already remarked, Moyal [16] starts by using the characteristicfunction defined in equation (2). He then constructs the expectation valueof a given operator, G(x, p) by means of the following equation,

G =! !

G(x, p)F (x, p)dxdp (8)

where G(x, p) is obtained from the operator form of G by first using theWeyl procedure [21] viz.

G(x, p) =! !

'(#, ")ei("x+! p)d#d" (9)

and then forming

G(x, p) =! !

'(#, ")ei("x+!p)d#d" (10)

To make the connection with Bohm’s formalism [8], we must followMoyal’s procedure of constructing the space-conditional moments of the mo-mentum. These are written as p and defined through

&(x)pn(x) =!

pnF (x, p)dp (11)

7

Using the expression for F (x, p) defined in equation (3) this can then bewritten in the form

&(x)pn(x) =$

!2i

% &$%

%x1! %

%x2

%n

!(x1)!!(x2)'

x1=x2=x

(12)

where &(x) =(

F (x, p)dp = !!(x)!(x). If we now write

!(x) = &1/2(x)eiS(x)/! (13)

we find for n = 1p = "S (14)

This is equation (A1.6) in the appendix of the paper by Moyal [16]. It isjust the momentum defined in Bohm above through equation (5) .

4 The Common Transport Equations

4.1 Transport of Probability

To discuss transport equations we need to know how the quasi-probabilitydistribution F (x, p, t) evolves in time. The simplest way to present this isto follow Moyal and introduce a new product, the star product, defined by

A(x, p) #B(x, p) := A(x, p)exp

)i!2

*$!%

%x

!%%

%p!!%%

%x

$!%

%p

+,B(x, p) (15)

Using this product we now introduce the Moyal bracket defined by

{A(x, p), B(x, p)}MB =A #B !B #A

i! (16)

The star product sets up what is known as the deformed Poisson algebra, socalled because if the Moyal bracket is expanded up to order (!), it becomesthe classical Poisson bracket. Substituting equation (15) into equation (16)shows that the Moyal bracket can be written as

{A(x, p), B(x, p)}MB = A(x, p)$

2!

%sin

)!2

*$!%

%x

!%%

%p!!%%

%x

$!%

%p

+,B(x, p)

(17)Then the time development of the quasi-probability distribution can be writ-ten in the form

%F (x, p)%t

+ {F (x, p), H(x, p)}MB = 0 (18)

8

In the limit to order !, this equation becomes the classical Liouville equation.If we write the Hamiltonian as H(x, p) = p2/2m + V (x) this equation

can be written in the form [22]

%F (x, p, t)%t

+p

m."F (x, p, t) =

!J(x, p! p$)F (p$, x, t)dp$ (19)

where

J(x, p! p$) = !i

![V (x! y/2)! V (x + y/2)]ei(p#p")dy (20)

This can also be written in the form

J(x, p) = !2!

1(2$!)3

!V (x + y/2) sin

"py

!

#dy

If we integrate equation (19) over p we find the RHS of equation (20) vanishesleaving

%&

%t+".

" &

m"S

#= 0 (21)

which is just the equation for the conservation of probability. This equationappears as equation (A4.2) in Moyal’s paper [16] and is identical to theBohm equation (6).

4.2 Transport of p

.In order to find the transport equation for p, we must multiply equation

(19) by pk to obtain

%pkF (x, p, t)%t

+-

i

pkpi

m

%F (x, p, t)%xi

=!

pkJ(x, p! p$)F (x, p$, t)dp$ (22)

Then integrating over p we find the RHS of equation (19) reduces to!&%V/%xk

so that the equation becomes

%(&pk)%t

+1m

-

i

%

%xi(&pipk) = !&

%V

%xk(23)

We can also show that the dispersion in momentum becomes

1m

-

i

%

%xi

.(&pipk)! (&pi.pk)

/= ! !2

4m

-

i

%

%xi

&&

%2 ln &

%xi%xk

'(24)

9

We can now use this result in equation (23) so that it can be written in theform

%(&pk)%t

+1m

-

i

%(&pi.pk)%xk

= !&%V

%xk! !2

4m

-

i

%

%xi

&&

%2 ln &

%xi%xk

'(25)

Di!erentiating the first term in equation (25), using equation (14) wefind

&%

%xk

)%S

%t+

12m

-

i

$%S

%xi

%2

+ V

,=

!2

4m

-

i

%

%xi

&&

%2 ln &

%xi%xk

'(26)

In order to bring the RHS of equation (26) into recognisable form let uswrite & = R2. Then it is straight forward to show

14m

-

i

%

%xi

&&

%2 ln &

%xi%xk

'=

12m

&%

%xk

-

i

&%2R

%x2i

/R

'(27)

so that equation (25) becomes

&%

%xk

&%S

%t+

12m

("S)2 + V ! !2

2m"2R/R

'= 0 (28)

Thus we immediately see the connection with the Bohm formalism as ex-pressed through equation (4). Equation (28) is essentially equation (A4.4) in Moyal’s paper [16]. His equation is derived for a charged parti-cle moving in an electromagnetic field using the more general HamiltonianH(pi, xi) = 1

2m

0i

1pi ! e

cAi22 + V (xi) and reduces to equation (28) in the

absence of the vector potential. Thus the three essential equations thatform the basis of the Bohm interpretation already appear in the appendixof Moyal’s classic paper.

In passing it should be noted that by combining equations (24) and (27)we have

pipk ! pi.pk =!2

2%

%xk

&%2R

%x2i

/R

'(29)

Thus we see that, mathematically, the quantum potential arises from dif-ference between the mean of the square of the momentum and the meanmomentum squared. All this means that the dispersion in the momentumfor a single particle in quantum mechanics can be nonzero. For the single

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particle in classical physics the momentum is always dispersion free. Equa-tion (29) shows that this di!erence can be written as the gradient of aquantity that has become known as the quantum potential.

The connection with the quantum potential is made even clearer whenone realises that the LHS of equation (25) is the total derivative of meanmomentum p, so that using equation (27) in equation (25) we find

dv

dt= !"[V + Q] (30)

where Q is the quantum potential Q = ! !2

2m"2R/R. Equation (30) explains

the origin of the name.

5 Conclusions.

What we have brought out in this paper is the mathematical relationshipbetween the formalisms used by Moyal and by Bohm. Both introduce thesame momentum PB = p and both produce the same quantum Hamilton-Jacobi equation (2). Although this relationship has already been pointedout by Takabayasi [22], it seems to have been forgotten in the present debateconcerning the interpretations of the quantum formalism. Take for examplethe excellent review article by Carruthers and Zachariasen [23] which dealsspecifically with approaches based on quantum mechanical phase-space dis-tributions. The original Wigner-Moyal work and its subsequent develop-ment is discussed in great detail, but the Bohm approach does not even geta mention. Yet the trajectories that play such a prominent role in the Bohmapproach are none other than the stream lines of the mean momentum fieldp(x, t) of the Moyal approach.

The reluctance to see the Bohm approach as an alternative representa-tion of the quantum formalism has always puzzled me [20]. I have alwaysseen it as a way of gaining a di!erent insight into quantum phenomena. Thereluctance to consider it seriously appears to come from the firm belief thatthe Heisenberg uncertainty principle ‘forbids’ us to attribute a simultaneousprecise position and momentum to an individual quantum particle. Howeversuch a strong assertion cannot be justified by appealing to the uncertaintyprinciple alone. What the uncertainty principle states is that it is not pos-sible to measure simultaneously the precise position and momentum of sucha particle, a feature that is built into the Bohm approach. Indeed that theHeisenberg uncertainty principle is about measurement has been backed upby many detailed analysis of experiments, both actual and hypothetical,

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that attempt to perform such a measurement. All fail, so this principle is inno doubt but in asserting this fact but one should note that it is all to dowith measurement.

What can we conclude from this fact? It seems to me we can drawone of two conclusions. Either we can assume the principle implies that aquantum particle cannot possibly have a precise position and momentum.Alternatively we can simply assume that a quantum particle can have aprecise position and momentum but we cannot measure and hence know thevalues of these variables simultaneously. Experiment cannot help to decidebetween these two possibilities because there is no actual experiment thatwill do it for us. In other words there is no experiment to directly showthat a quantum particle cannot have precise values for these two variables.Equally there is no experiment to directly show that a quantum particle canhave simultaneous position and momentum. Thus both the standard andphase-space approaches depend crucially on which assumption is adopted atthis point.

Since the original assumption used by the Bohm approach cannot betested directly, there has always been the hope that there may be someindirect experimental way to distinguish this approach from the standardapproach. But how can there be any di!erence? Both the standard andphase-space approaches use exactly the same mathematical structure.

The Wigner-Moyal approach derives all its results starting from the two-point density matrix equation. On the other hand the Bohm approachderives its results directly from the Schrodinger equation. Neither add anynew mathematical content. Both of these approaches are designed to giveexactly the same statistical results as the standard formalism therefore therecannot be any measurable di!erence. Therein lies the weakness of proposingthese approaches as ‘new’ theories; they are not new theories. They aresimply di!erent ways of looking at the same formalism.

The question then is why should we explore these di!erent approaches?They predict no new results that could not be deduced from the standardapproach so why bother? One answer has already been outlined in theintroduction. Once we have a di!erent way of looking at the quantumformalism, we begin to see quantum phenomena in a di!erent perspectiveso that we open up new possibilities. The fact that we can use the conceptof a trajectory for a quantum particle suggests that in attempting to finda quantum theory of gravity, we do not have to construct a Hilbert spacebut can try to see how the geodesics of the classical theory may be modifiedto produce quantum behaviour. Of course I am not saying that, therefore,we have a quantum theory of gravity, far from it, but at least we have the

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possibility of using a similar mathematical language to discuss both topics.Of course there is a more direct answer to the ‘why bother’ question.

The standard approach giving rise to the well known apparently inexpli-cable paradoxes, particle behaviour in two-slit interference experiments,schizophrenic cats, delayed choice paradox, quantum erasure and the mea-surement problem, which lead to so many conceptual di"culties. The mea-surement problem in particular consumes an immense amount of intellectualenergy and has generated a vast literature seemingly without and end insight. The Bohm approach has none of these di"culties.

For me the discussion of the measurement problem in the standard ap-proach always hints at a degree of contradiction. The original assumption indenying that a quantum particle has a simultaneous position and momentumwas supported by what I can only describe as the ‘feeling’ that we shouldnot be using properties that cannot be directly measured, fitting in with theprevailing mood of positivism at the time the standard interpretation wasbeing established. But this assumption seems to have been abandoned whenone decides to look for ways to “collapse the wave function”. It is generallyassumed that the wave function cannot be measured directly, yet in thiswhole discussion, we are implicitly treating it as something that is physicalor “real” that has to “collapse”. It is assumed to be part of the physicalprocess even though it cannot be measured. Contrast that with the strongresistance to attribute simultaneous values of position and momentum to aparticle.

All of this is not to imply that the phase-space approaches supply us withall the answers. The phase space that is constructed is not a classical spaceand on further investigation shows some interesting deeper structure. Forexample, as we have already remarked above, equation (19) can be obtaineddirectly from a two-point density matrix & = !!(x)!(x$). Changing variablesto X = (x + x$)/2 and y = x! x$, we find

&(X ! !y/2, X + !y/2) =!

F (X,P )e#iyP/!dP

Thus the (X, P ) used in the Wigner distribution are the mean values ofsome form of cell structure in phase space, which hints at a deeper un-derlying topological structure [26]. de Gosson [27] also discusses these cellstructures, calling them “quantum blobs”. He shows that they are relatedto the topological notion of capacity, a notion that arises naturally in sym-plectic geometry. It is also interesting to note that F (X, P ) turns out tobe identical to the ambiguity function used extensively in radar theory [24][25] suggesting that we are looking at the average behaviour of some form of

13

underlying cellular dynamics. All this suggests that some form of underlyingsmall scale topology which could fluctuate, resembling the foam-like struc-ture suggested by Wheeler in his proposals of what he calls “pre-geometry”[4].

Some preliminary results of my own based on this type of topologicalambiguity has been reported elsewhere [26]. However even this preliminarywork suggests that a more radical approach based on non-commutative ge-ometry [29] or more radically still on a form of non-commutative topologyalong the lines suggested by van Oystaeyen [28]. These ideas will be discusselsewhere.

6 Acknowledgements

I should like to thank Professor Robert Rynasiewicz for inviting me topresent some of these ideas at his conference “New Directions in the Foun-dations of Physics” held at the University of Maryland in April 2007.

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[28] F. van Oystaeyen, Virtual Topology and Functor Geometry To be pub-lished 2006.

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